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5.2.3 Recall

In recall, the components of a pattern $ \bf p$ are partially given, and the components of a completed pattern $ \bf z$ are divided into input and output (see section 4.2). The input is the offset $ \bf p$ of a hyper-plane extending into the output dimensions:

$\displaystyle \bf z$ = $\displaystyle \bf M$$\displaystyle \eta$ + $\displaystyle \bf p$ , (5.10)

where $ \eta$ are the free parameters of the hyper-plane. The matrix $ \bf M$ defines which dimensions are input and which are output. To obtain the output for a given input, the free parameters are chosen such that the potential E($ \bf z$) is minimized,


$\displaystyle \eta$* = arg$\displaystyle \min_{{\boldsymbol{\eta}}}^{}$E($\displaystyle \bf z$($\displaystyle \eta$)) ,  
$\displaystyle \bf z_{{\mbox{\footnotesize opt}}}^{}$ = $\displaystyle \bf M$$\displaystyle \eta$* + $\displaystyle \bf p$ . (5.11)

There seems to be no analytical solution for $ \eta$*. In feature space, the constraint space is not a hyper-plane. However, the minimization can be solved using standard numerical optimization algorithms.


next up previous contents
Next: 5.3 Experiments Up: 5.2 Pattern association algorithm Previous: 5.2.2.1 Examples
Heiko Hoffmann
2005-03-22